)-hook formula for tailed insets and a macdonald ...mi/comb2018/talks/feb23/ishikawa201… ·...

. . . . . .

......

(q, t)-hook formula for Tailed Insets and aMacdonald polynomial identity

Masao Ishikawa†

†Okayama University

Algebraic and Enumerative Combinatorics in OkayamaFebruary 23, 2018

Okayama University

Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . . . .

.. Abstract

.Abstract..

......

Okada presented a conjecture on (q, t)-hook formula for generald-complete posets in the paper, Soichi Okada, (q, t)-Deformationsof multivariate hook product formulae, J. Algebr. Comb. (2010) 32,399 – 416. We consider the Tailed Inset case, and reduce theconjectured identity to an indentity of the Macodonald polynomialsrephrasing Okada’s (q,t)-weights via Pieri coefficients of theMacodonald polynomials. Joint work with Frederic Jouhet(University of Lyon I).

. . . . . .

.. References

In this talk...1 M. Ishikawa, “(q, t)-hook formula for Birds and Banners”,arXiv:1302.1968 [math.CO].

...2 S. Okada, “(q, t)-deformations of multivariate hook productformulae”, arXiv:0909.0086 [math.CO] J. AlgebraicCombin. 32 (2010), 399-416.

...3 R. Proctor, “Dynkin diagram classication of λ-minusule Bruhatlattices and of d-complete posets”, J. Algebraic Combin. 9(1999), 61 – 94.

...4 M. Vuletic, “A generalization of Macmahon’s formula”,arXiv:0707.0532 [math.CO] 4Jul 2007, Trans. Amer.Math. Soc. 361 (2009), 2789-2804.

...5 S.O. Warnaar, “Rogers-Szego polynomials andHall-Littlewood symmetric functions”, J. Algebra 303 (2006),810–830.

. . . . . .

.. References

. . . . . .

.. References

. . . . . .

.. References

. . . . . .

.. References

. . . . . .

.. Additional References

Further references:...6 G. Gasper, “Rogers’ linearization formula for the continuous

q-ultraspherical polynomials and quadratic transformationformulas”, SIAM J. Math. Anal., 16 (1985), 1061-1071.

...7 I. G. Macdonald, Symmetric Functions and Hall Polynomials(2nd ed.), Oxford Univ. Press, (1995).

...8 G. Gasper and M. Rahman, Basic Hypergeometric Series(2nd ed.), Cambridge Univ. Press, (1990, 2004).

...9 R. Stanley, Enumerative Combinatorics: Volume 1, 2 (2nded.), Cambridge Univ. Press, (2001, 2012).

...10 R. Stanley, Ordered Structures and Partitions, Memoirs AMSno. 119 (1972).

. . . . . .

Introduction

. . . . . .

.. P-partitions.Definition..

......

A partially ordered set (also called a poset) is a set P with a binaryrelation “≤” which is antisymmetric, transitive, and reflexive.

.Definition (Stanley ’72)..

......

Let P be a poset. A P-partition is a map π : P → N satisfying

x ≤ y in P =⇒ π(x) ≥ π(y) in N,

where N is the set of nonnegative integers. Let A (P) be the set ofP-partitions.

.Example (P-partitions)..

......

. . . . . .

......

x ≤ y in P =⇒ π(x) ≥ π(y) in N,

......

. . . . . .

......

x ≤ y in P =⇒ π(x) ≥ π(y) in N,

......

. . . . . .

......

x ≤ y in P =⇒ π(x) ≥ π(y) in N,

......

. . . . . .

......

x ≤ y in P =⇒ π(x) ≥ π(y) in N,

......

. . . . . .

......

x ≤ y in P =⇒ π(x) ≥ π(y) in N,

......

. . . . . .

......

x ≤ y in P =⇒ π(x) ≥ π(y) in N,

......

. . . . . .

......

x ≤ y in P =⇒ π(x) ≥ π(y) in N,

......

. . . . . .

.. (Shifted) diagrams.Definition..

......

A partiton is a nonincreasing sequence λ = (λ1, λ2, . . . ) of nonnegativeintegers with finitely many λi unequal to zero. The length and weight ofλ, denoted by ℓ(λ) and |λ|, are the number and sum of the non-zero λi

respectively. A strict partition is a partition in which its parts are strictlydecreasing. If λ is a partition (resp. strict partition), then its diagramD(λ) (resp. shifted diagram S(λ)) is defined by

D(λ) = { (i, j) ∈ Z2 : 1 ≤ j ≤ λi}S(λ) = { (i, j) ∈ Z2 : i ≤ j ≤ λi + i − 1 }.

.Example (The diagram and shifted diagram for λ = (4, 3, 1))..

......

D(λ) = S(λ) =

. . . . . .

......

D(λ) = S(λ) =

. . . . . .

......

D(λ) = S(λ) =

. . . . . .

......

D(λ) = S(λ) =

. . . . . .

......

D(λ) = S(λ) =

. . . . . .

.. (Shifted) shapes.Definition..

......

A diagram D(λ) or a shifted diagram S(λ) is regarded as a posetby defining its order structure by

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

By this order the poset represented by a diagram P = D(λ) iscalled a shape, and the posets P = S(λ) is called shifted shapes.

.Example (The shape and shifted shape for λ = (4, 3, 1))..

......

D(λ) = S(λ) =

. . . . . .

......

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

......

D(λ) = S(λ) =

......

D(λ) = S(λ) =

. . . . . .

......

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

......

D(λ) = S(λ) =

. . . . . .

.. Hook.Definition..

......

For a partition (resp. strict partition) λ and a cell (i, j) ∈ D(λ) (resp.S(λ)), the hook at (i, j) in D(λ) (resp. S(λ)), is defined by

HD(λ)(i, j) = {(i, j)} ∪ {(i, l) ∈ D(λ) : l > j} ∪ {(k , j) ∈ D(λ) : k > i}

(resp.

HS(λ)(i, j) = {(i, j)} ∪ {(i, l) ∈ S(λ) : l > j}∪ {(k , j) ∈ D(λ) : k > i} ∪ {(j + 1, l) ∈ S(λ) : l > j}).

.Example (The hook at (1, 2) in D(λ) and S(λ) for λ = (4, 3, 1))..

......

D(λ) = S(λ) =

. . . . . .

.. Hook.Definition..

......

For a partition (resp. strict partition) λ and a cell (i, j) ∈ D(λ) (resp.S(λ)), the hook at (i, j) in D(λ) (resp. S(λ)), is defined by

HD(λ)(i, j) = {(i, j)} ∪ {(i, l) ∈ D(λ) : l > j} ∪ {(k , j) ∈ D(λ) : k > i}

(resp.

HS(λ)(i, j) = {(i, j)} ∪ {(i, l) ∈ S(λ) : l > j}∪ {(k , j) ∈ D(λ) : k > i} ∪ {(j + 1, l) ∈ S(λ) : l > j}).

.Example (The hook at (1, 2) in D(λ) and S(λ) for λ = (4, 3, 1))..

......

D(λ) = S(λ) =

. . . . . .

.. Content and hook length.Definition..

......

The hook length at (i, j) is defined by hD(λ)(i, j) = |HD(λ)(i, j)| (resp.hS(λ)(i, j) = |HS(λ)(i, j)|). Further c(i, j) = j − i is called the contentat (i, j).

.Example (The hook lenghs in D(λ) and S(λ) for λ = (4, 3, 1))..

......

D(λ) = S(λ) =

6 4 3 14 2 11

7 5 4 24 3 1

.Example (The contents in D(λ) and S(λ) for λ = (4, 3, 1))..

......

D(λ) = S(λ) =

0 1 2 3−1 0 1−2

0 1 2 30 1 2

. . . . . .

......

D(λ) = S(λ) =

6 4 3 14 2 11

7 5 4 24 3 1

......

D(λ) = S(λ) =

0 1 2 3−1 0 1−2

0 1 2 30 1 2

. . . . . .

......

D(λ) = S(λ) =

6 4 3 14 2 11

7 5 4 24 3 1

......

D(λ) = S(λ) =

0 1 2 3−1 0 1−2

0 1 2 30 1 2

. . . . . .

.. One Variable Hook Length Formula

.Theorem (Frame-Robinson-Thrall ’54, Stanley ’72))..

......

If P = D(λ) or S(λ), then we have∑π∈A (P)

z |π| =∏(i,j)∈P

1 − zhP(i,j),

where the sum on the left-hand side runs over all P-partitions, and|π| = ∑

x∈P π(x).

.Example (An example of P-partition)..

......

π =|π| = 16

z |π| = z16

0 0 1 22 3 44

. . . . . .

.. One Variable Hook Length Formula

.Theorem (Frame-Robinson-Thrall ’54, Stanley ’72))..

......

If P = D(λ) or S(λ), then we have∑π∈A (P)

z |π| =∏(i,j)∈P

1 − zhP(i,j),

where the sum on the left-hand side runs over all P-partitions, and|π| = ∑

x∈P π(x).

......

π =|π| = 16

z |π| = z16

0 0 1 22 3 44

. . . . . .

.. Example of One Variable Hook Length Formula

.Example (The shape for λ = (4, 3, 1))..

......

D(λ) = D(λ) =

π11π12π13π14

π21π22π23

6 4 3 14 2 11

∑π∈A (D(λ))

(i,j)∈D(λ) πi,j =1

(1 − z)3(1 − z2)(1 − z3)(1 − z4)2(1 − z6).

. . . . . .

.. Multivariate Hook Length Formula

.Theorem (Gansner ’81, Sagan ’82)..

......

Let . . . , z−1, z0, z1, z2, . . . be variables. If P = D(λ) or S(λ), thenwe have ∑

π∈A (P)

zπ =∏(i,j)∈P

11 − z[HP(i, j)]

where the sum on the left-hand side runs over all P-partitions,zπ =

∏(i,j)∈P zc(i,j)

πi,j and z[H] =∏

(i,j)∈H zc(i,j) for any finite subsetH ⊂ Z2. (Gansner used Hillman-Grassl ’76 algorithm.)

......

π =zπ = z4

−2z2−1z3

0z41z2z2

0 0 1 22 3 44

. . . . . .

.. Multivariate Hook Length Formula

.Theorem (Gansner ’81, Sagan ’82)..

......

Let . . . , z−1, z0, z1, z2, . . . be variables. If P = D(λ) or S(λ), thenwe have ∑

π∈A (P)

zπ =∏(i,j)∈P

11 − z[HP(i, j)]

where the sum on the left-hand side runs over all P-partitions,zπ =

∏(i,j)∈P zc(i,j)

πi,j and z[H] =∏

(i,j)∈H zc(i,j) for any finite subsetH ⊂ Z2. (Gansner used Hillman-Grassl ’76 algorithm.)

......

π =zπ = z4

−2z2−1z3

0z41z2z2

0 0 1 22 3 44

. . . . . .

.. Example of Multivariate Hook Length Formula

......

D(λ) = D(λ) =

π11π12π13π14

π21π22π23

z0 z1 z2 z3

z−1 z0 z1

∑π∈A (P)

zπ31−2 zπ21

−1 zπ11+π220 zπ12+π23

1 zπ132 zπ14

(1 − z−2z−1z0z1z2z3)(1 − z0z1z2z3)(1 − z1z2z3)(1 − z3)

× 1(1 − z−2z−1z0z1)(1 − z0z1)(1 − z1)(1 − z−2)

. . . . . .

.. The Cauchy formula and the Littlewood formula

.Therem (The Cauchy formula)..

......

Let x = (x1, . . . , xn) and y = (y1, . . . , yn) are n-tuples of variables.Then we have ∑

sλ(x)sλ(y) =n∏

11 − xiyj

.Therem (The Littlewood formula)..

......

Let x = (x1, . . . , xn) is an n-tuples of variables. Then we have

sλ(x) =n∏

11 − xi

∏1≤i<j≤n

11 − xixj

. . . . . .

.. The Cauchy formula and the Littlewood formula

.Therem (The Cauchy formula)..

......

Let x = (x1, . . . , xn) and y = (y1, . . . , yn) are n-tuples of variables.Then we have ∑

sλ(x)sλ(y) =n∏

11 − xiyj

.Therem (The Littlewood formula)..

......

Let x = (x1, . . . , xn) is an n-tuples of variables. Then we have

sλ(x) =n∏

11 − xi

∏1≤i<j≤n

11 − xixj

. . . . . .

.. (q, t)-hook formula.Conjecture (Okada ’10)..

......

If P is a d-complete poset, then we have∑π∈A (P)

WP(π; q, t) zπ =∏(i,j)∈P

F (z[HP(i, j)]; q, t) ,

where the sum on the left-hand side runs over all P-partitions, and

F(x; q, t) =(tx; q)∞(x; q)∞

......

D(λ) = D(λ) =

π11π12π13π14

π21π22π23

z0 z1 z2 z3

z−1 z0 z1

∑π∈A (P)

WP(π; q, t) zπ31−2 zπ21

−1 zπ11+π220 zπ12+π23

1 zπ132 zπ14

= F (z−2z−1z0z1z2z3; q, t)F (z0z1z2z3; q, t)F (z1z2z3; q, t)

× F (z3; q, t)F (z−2z−1z0z1; q, t)F (z0z1; q, t)F (z1; q, t)F (z−2; q, t) .

. . . . . .

.. (q, t)-hook formula

......

D(λ) = D(λ) =

π11π12π13π14

π21π22π23

z0 z1 z2 z3

z−1 z0 z1

∑π∈A (P)

WP(π; q, t) zπ31−2 zπ21

−1 zπ11+π220 zπ12+π23

1 zπ132 zπ14

= F (z−2z−1z0z1z2z3; q, t)F (z0z1z2z3; q, t)F (z1z2z3; q, t)

× F (z3; q, t)F (z−2z−1z0z1; q, t)F (z0z1; q, t)F (z1; q, t)F (z−2; q, t) .

. . . . . .

.. Current situation

.Current situation..

......

...1 If P is (1) Shape or (2) Shfted Shape, the (q, t)-hook formulais proven in the paper by Okada(2010).

...2 If P is (3) Bird or (6) Banner, the (q, t)-hook formula is provenby me (not yet published) 2013. We use Gasper’s identity.

...3 This talk is about the Tailed Inset case (not yet completed).

. . . . . .

......

. . . . . .

......

. . . . . .

......

. . . . . .

.. The Cauchy type identity for Macdonald polynomials

.Theorem..

......

Let x = (x1, . . . , xn) and y = (y1, . . . , yn) are n-tuples of variables.Then we have∑

Pλ(x; q, t)Qλ(y; q, t) =n∏

F(xiyj; q, t).

. . . . . .

.. Warnaar’s formula

.Theorem (Warnaar ’06)..

......

w r(λ)boaλ (q, t)Pλ(x; q, t) =

∏i≥1

(1 + wxi)(qtx2i ; q

(x2i ; q

∏i<j

(txixj ; q)∞(xixj ; q)∞

where r(λ) is the number of rows of odd length.

.Further Corollary..

......

w|λ|+r(λ′)

2 belλ (q, t)Pλ(x; q, t) =

∏i≥1

(twxi ; q)∞(wxi ; q)∞

∏i<j

(twxixj ; q)∞(wxixj ; q)∞

w|λ|−r(λ′)

∏i≥1

(txi ; q)∞(xi ; q)∞

∏i<j

. . . . . .

.. Warnaar’s formula.Theorem (Warnaar ’06)..

......

∏i≥1

(1 + wxi)(qtx2i ; q

(x2i ; q

∏i<j

.Corollary..

......

w r(λ′)belλ (q, t)Pλ(x; q, t) =

∏i≥1

∏i<j

Proof. Applying the F-algebra homomorphism wq,t to the aboveidentity..Further Corollary..

......

w|λ|+r(λ′)

∏i≥1

∏i<j

w|λ|−r(λ′)

∏i≥1

(txi ; q)∞(xi ; q)∞

∏i<j

. . . . . .

.. Warnaar’s formula.Theorem (Warnaar ’06)..

......

∏i≥1

(1 + wxi)(qtx2i ; q

(x2i ; q

∏i<j

.Corollary..

......

w r(λ′)belλ (q, t)Pλ(x; q, t) =

∏i≥1

∏i<j

Proof. Applying the F-algebra homomorphism wq,t to the aboveidentity..Further Corollary..

......

w|λ|+r(λ′)

∏i≥1

∏i<j

w|λ|−r(λ′)

∏i≥1

(txi ; q)∞(xi ; q)∞

∏i<j

. . . . . .

.. Warnaar’s formula

.Theorem (Warnaar ’06)..

......

∏i≥1

(1 + wxi)(qtx2i ; q

(x2i ; q

∏i<j

.Further Corollary..

......

w|λ|+r(λ′)

∏i≥1

∏i<j

w|λ|−r(λ′)

∏i≥1

(txi ; q)∞(xi ; q)∞

∏i<j

. . . . . .

d-complete poset

. . . . . .

.. d-complete poset

.Contents of this section..

......

...1 The d-complete posets arise from the dominant minusculeheaps of the Weyl groups of simply-laced Kac-Moody Liealgebras.

...2 Proctor gave completely combinatorial description ofd-complete poset, which is a graded poset with d-completecoloring.

...3 Proctor showed that any d-complete poset can be obtainedfrom the 15 irreducible classes by slant-sum.

...4 The d-complete coloring is important for the multivariategenerating function. The content should be replaced by colorfor d-complete posets.

...5 Okada’s (q, t)-weight WP(π; q, t)

...6 Hook monomials for d-complete posets

. . . . . .

.. d-complete poset

......

. . . . . .

.. d-complete poset

......

. . . . . .

.. d-complete poset

......

. . . . . .

.. d-complete poset

......

. . . . . .

.. d-complete poset

......

. . . . . .

.. d-complete poset

......

. . . . . .

.. Double-tailed diamond poset

.Definition..

......

The double-tailed diamond poset dk (1) is the poset depictedbelow:

k − 2

side side

bottom

A dk -interval is an interval isomorphic to dk (1).

A d−k -interval (k ≥ 4) is an interval isomorphic to dk (1) − {top}.A d−3 -interval consists of three elements x, y and w such thatw is covered by x and y.

. . . . . .

.Definition..

......

k − 2

side side

bottom

. . . . . .

.Definition..

......

k − 2

side side

bottom

. . . . . .

.Definition..

......

k − 2

side side

bottom

. . . . . .

.Definition..

......

k − 2

side side

bottom

. . . . . .

.. Definition of d-complete poset

.Definition..

......

A poset P is d-complete if it satisfies the following three conditionsfor every k ≥ 3:

...1 If I is a d−k -interval, then there exists an element v such that vcovers the maximal elements of I and I ∪ {v} is a dk -interval.

...2 If I = [w, v] is a dk -interval and the top v covers u in P, thenu ∈ I.

...3 There are no d−k -intervals which differ only in the minimalelements.

. . . . . .

.Definition..

......

. . . . . .

.Definition..

......

. . . . . .

.Definition..

......

. . . . . .

.. Properties of d-complete posets

.Fact..

......

If P is a connected d-complete poset, then

(a) P has a unique maximal element.

(b) P is graded, i.e., there exists a rank function r : P → N suchthat r(x) = r(y) + 1 if x covers y.

.Fact..

......

(a) Any connected d-complete poset is uniquely decomposed intoa slant sum of one-element posets and slant-irreducibled-complete posets.

(b) Slant-irreducible d-complete posets are classified into 15families : (1) Shapes, (2) Shifted shapes, (3) Birds, (4) Insets,(5) Tailed insets, (6) Banners, (7) Nooks, (8) Swivels, (9)Tailed swivels, (10) Tagged swivels, (11) Swivel shifts, (12)Pumps, (13) Tailed pumps, (14) Near bats, (15) Bat.

. . . . . .

.Fact..

......

.Fact..

......

. . . . . .

.Fact..

......

.Fact..

......

. . . . . .

.Fact..

......

.Fact..

......

. . . . . .

.Fact..

......

.Fact..

......

. . . . . .

.Fact..

......

.Fact..

......

. . . . . .

.. Examples

rooted tree

shifted shape

swivel

. . . . . .

.. 15 irreducible d-complete posets

. . . . . .

.. Top Tree and d-Complete Coloring.Definition..

......

For a connected d-complete poset P, we define its top tree byputting

T = {x ∈ P : every y ≥ x is covered by at most one other element }

.Fact..

......

Let I be a set of colors such that #I = #T . Then a bijectionc : T → I can be uniquely extended to a map c : P → I satisfyingthe following three conditions:

If x and y are incomparable, then c(x) , c(y).

If an interval [w, v] is a chain, then the colors c(x) (x ∈ [w, v])are distinct.

If [w, v] is a dk -interval then c(w) = c(v).

Such a map c : P → I is called a d-complete coloring.

. . . . . .

......

.Fact..

......

. . . . . .

......

.Fact..

......

. . . . . .

......

.Fact..

......

. . . . . .

......

.Fact..

......

. . . . . .

.. Top Tree and d-Complete Coloring

.Example..

......

Top Tree and d-Complete Coloring of d5-interval

. . . . . .

.Example..

......

. . . . . .

.Example..

......

. . . . . .

.Example..

......

. . . . . .

.. Shapes

.Definition..

......

Let r be a postive integer, and α = (α1, . . . , αr) andβ = (β1, . . . , βr) be strict partitions such that

α1 > · · · > αr ≥ 0, β1 > · · · > βr ≥ 0,

Let P be the set P = PL ∪ PR of lattice points in Z2, where

PR = { (i, j) : 1 ≤ i ≤ j ≤ αi + i − 1 (1 ≤ i ≤ r) },PL = { (i, j) : 1 ≤ j ≤ i ≤ βj + j − 1 (1 ≤ j eqr) },

We regard P as a poset by defining the order relation

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

We call this poset a shape and denote it by P = P1(α, β).

. . . . . .

.. Shapes

.Definition..

......

α1 > · · · > αr ≥ 0, β1 > · · · > βr ≥ 0,

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Shapes

.Definition..

......

α1 > · · · > αr ≥ 0, β1 > · · · > βr ≥ 0,

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Shapes

. . . . . .

.. Shifted Shapes

.Definition..

......

Let r be a postive integer, and α = (α1, . . . , αr) be a strict partitionsuch that

α1 > · · · > αr ≥ 0.

Define the shifted shape P = P2(α) by

P = { (i, j) : i ≤ j ≤ αi + i − 1 (1 ≤ i ≤ r) }.

We regard it as a poset by defining its order structure

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Shifted Shapes

.Definition..

......

α1 > · · · > αr ≥ 0.

P = { (i, j) : i ≤ j ≤ αi + i − 1 (1 ≤ i ≤ r) }.

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Shifted Shapes

.Definition..

......

α1 > · · · > αr ≥ 0.

P = { (i, j) : i ≤ j ≤ αi + i − 1 (1 ≤ i ≤ r) }.

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Shifted Shape

. . . . . .

.. Birds.Definition..

......

Let α = (α1, α2) and β = (β1, β2) be strict partitions such that α1 > α2 > 0and β1 > β2 > 0. Define the bird P = P3(α, β; f) by

P = PH ∪ PR ∪ PL ∪ PT

PH = { (1, j) : −f + 1 ≤ j ≤ 1},PR = { (i, j) : i ≤ j ≤ αi + i − 1 (i = 1, 2) },PL = { (i, j) : j ≤ i ≤ βj + j − 1 (j = 1, 2) },PT = { (i, i) : 2 ≤ i ≤ f + 2}

as a set and we regard it as a poset by defining its order structure

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

if and only if the both of (i1, j1) and (i2, j2) are in PH ∪ PR ∪ PL or in PT

. . . . . .

......

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

......

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Birds

. . . . . .

.. Tailed Insets

.Definition..

......

Let α = (α1, α2, α3) and β = (β1, β2) be strict partitions such that

α1 > α2 > α3 ≥ 0, β1 > β2 ≥ 0.

Let P be the set P = PH ∪ PM ∪ PL ∪ PR ∪ PT of lattice points in Z2,where PM = { (2, 1) }, PT = { (4, 4) } and

PH = { (1, j) : −β1 + 1 ≤ j ≤ 0},PR = { (i, j) : 1 ≤ i ≤ j ≤ αi + i (i = 1, 2, 3) },PL = { (i + 1, j + 1) : 1 ≤ j ≤ i ≤ βj + j (j = 1, 2) }.

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

if neither of (i1, j1) and (i2, j2) is not in PT, whereas (3, 3) < (4, 4). We callthis poset a Tailed Inset, denoted by P5(α, β).

. . . . . .

.. Tailed Insets

.Definition..

......

α1 > α2 > α3 ≥ 0, β1 > β2 ≥ 0.

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Tailed Insets

.Definition..

......

α1 > α2 > α3 ≥ 0, β1 > β2 ≥ 0.

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Tailed Insets

(1, 1)

. . . . . .

.. Banners

.Definition..

......

Let α = (α1, α2, α3, α4) be a strict partition such thatα1 > α2 > α3 > α4 > 0, and let f ≥ 2 be a positive integer. Let Pbe the set P = PH ∪ PW ∪ PT of lattice points in Z2, where

PH = { (1, j) : −f + 2 ≤ j ≤ 1},PW = { (i, j) : i ≤ j ≤ αi + i − 1 (i = 1, 2, 3, 4) },PT = { (i, 3) : 3 ≤ i ≤ f + 2}.

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

if both of (i1, j1) and (i2, j2) are in PH ∪ PW or in PT, and call it abanner

. . . . . .

.. Banners

.Definition..

......

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Banners

.Definition..

......

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Banners

vf v1 v

. . . . . .

.. Hook Monomials

.Definition..

......

Let P be a connected d-complete poset and T its top tree. Let zv

(v ∈ T ) be indeterminate. Let c : P → T be the d-completecoloring. For each v ∈ P, we define monomials z [HP(v)] byinduction as follows:

(a) If v is not the top of any dk -interval, then we deine

z [HP(v)] =∏w≤v

zc(w).

(b) If v is the top of a dk -interval [w, v], then we de?ne

z [HP(v)] =z [HP(x)] z [HP(Y)]

z [HP(w)]

where x and y are the sides of [w, v].

. . . . . .

.. Hook Monomials

.Definition..

......

z [HP(v)] =∏w≤v

zc(w).

z [HP(w)]

. . . . . .

.. Hook Monomials

.Definition..

......

z [HP(v)] =∏w≤v

zc(w).

z [HP(w)]

. . . . . .

.. An example of hook monomials

.Example..

......

We consider the following poset P = d1(5). We give the followingassignment of variavles for zc(x), x ∈ P.

. . . . . .

.. An example of hook monomials

.Example..

......

We consider the following poset P = d1(5). We give the followingassignment of variavles for zc(x), x ∈ P.

. . . . . .

.. Hook Monomials for P = d1(5)

.Example..

......

We consider the following poset P = d1(5). The monomialsassociated to hooks of P = d1(5) are as follows:

z1z22z2

z1z2z23z4z5

z1z2z3z4z5

z1z2z3

z1z2z3z4

z1z2z3z5

. . . . . .

.Example..

......

z1z22z2

z1z2z23z4z5

z1z2z3z4z5

z1z2z3

z1z2z3z4

z1z2z3z5

. . . . . .

.Example..

......

z1z22z2

z1z2z23z4z5

z1z2z3z4z5

z1z2z3

z1z2z3z4

z1z2z3z5

. . . . . .

.Example..

......

z1z22z2

z1z2z23z4z5

z1z2z3z4z5

z1z2z3

z1z2z3z4

z1z2z3z5

. . . . . .

.Example..

......

z1z22z2

z1z2z23z4z5

z1z2z3z4z5

z1z2z3

z1z2z3z4

z1z2z3z5

. . . . . .

.Example..

......

z1z22z2

z1z2z23z4z5

z1z2z3z4z5

z1z2z3

z1z2z3z4

z1z2z3z5

. . . . . .

.Example..

......

z1z22z2

z1z2z23z4z5

z1z2z3z4z5

z1z2z3

z1z2z3z4

z1z2z3z5

. . . . . .

.Example..

......

z1z22z2

z1z2z23z4z5

z1z2z3z4z5

z1z2z3

z1z2z3z4

z1z2z3z5

. . . . . .

.Example..

......

z1z22z2

z1z2z23z4z5

z1z2z3z4z5

z1z2z3

z1z2z3z4

z1z2z3z5

. . . . . .

.. (q, t)-Weight associated with P-partition π

.Definition..

......

Let P be a connected d-complete poset with top tree T . Given aP-partition π ∈ A (P), we define WP(π; q, t) by

∏x,y∈P, x<y

c(x) and c(y) are adjacent in T

f(π(x) − π(y),

⌊r(y)−r(x)

⌋)∏

x,y∈P, x<yc(x)=c(y)

f(π(x) − π(y),

⌊r(y)−r(x)

⌋)f(π(x) − π(y),

⌊r(y)−r(x)

⌋− 1

) ..Example........Compute this weight WP(π; q, t) for P = d1(5).

. . . . . .

.. (q, t)-Weight associated with P-partition π

.Definition..

......

Let P be a connected d-complete poset with top tree T . Given aP-partition π ∈ A (P), we define WP(π; q, t) by

∏x,y∈P, x<y

c(x) and c(y) are adjacent in T

f(π(x) − π(y),

⌊r(y)−r(x)

⌋)∏

x,y∈P, x<yc(x)=c(y)

f(π(x) − π(y),

⌊r(y)−r(x)

⌋)f(π(x) − π(y),

⌊r(y)−r(x)

⌋− 1

) ..Example........Compute this weight WP(π; q, t) for P = d1(5).

. . . . . .

.. Example

.Example..

......

We consider the following poset P = d5(1). A P-partition π mustsatisfy the following inequalities

π11 π12 π13 π14

π23 π24

≤ ≤ ≤≤≤

≥ ≥≥

≥Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . . . .

.. Example

.Example..

......

π11 π12 π13 π14

π23 π24

≤ ≤ ≤≤≤

≥ ≥≥

. . . . . .

.. Example

.Example..

......

π11 π12 π13 π14

π23 π24

≤ ≤ ≤≤≤

≥ ≥≥

. . . . . .

.. Numerator of WP(π; q, t) (P = d1(5)).Example..

......

A P-partition π extends to P-partition π.

0 π11 π12 π13 π14

π23 π24

numer WP(π; q, t) = f(π11; 0)f(π44; 3)

× f(π12 − π11; 0)f(π34 − π11; 2)f(π44 − π12; 2)f(π44 − π34; 0)

× f(π13 − π12; 0)f(π24 − π12; 1)f(π34 − π13; 1)f(π34 − π13; 0)

× f(π14 − π13; 0)f(π24 − π13; 0)f(π23 − π13; 0)f(π24 − π23; 0)

. . . . . .

......

0 π11 π12 π13 π14

π23 π24

numer WP(π; q, t) = f(π11; 0)f(π44; 3)

× f(π12 − π11; 0)f(π34 − π11; 2)f(π44 − π12; 2)f(π44 − π34; 0)

× f(π13 − π12; 0)f(π24 − π12; 1)f(π34 − π13; 1)f(π34 − π13; 0)

× f(π14 − π13; 0)f(π24 − π13; 0)f(π23 − π13; 0)f(π24 − π23; 0)

. . . . . .

......

0 π11 π12 π13 π14

π23 π24

numer WP(π; q, t) = f(π11; 0)f(π44; 3)

× f(π12 − π11; 0)f(π34 − π11; 2)f(π44 − π12; 2)f(π44 − π34; 0)

× f(π13 − π12; 0)f(π24 − π12; 1)f(π34 − π13; 1)f(π34 − π13; 0)

× f(π14 − π13; 0)f(π24 − π13; 0)f(π23 − π13; 0)f(π24 − π23; 0)

. . . . . .

.. Denominator of WP(π; q, t) (P = d1(5))

.Example..

......

A P-partition π

π11 π12 π13 π14

π23 π24

denom WP(π; q, t) = f(π44 − π11; 2)f(π44 − π11; 3)

× f(π34 − π12; 1)f(π34 − π12; 2)f(π24 − π13; 0)f(π24 − π13; 1)

. . . . . .

.Example..

......

A P-partition π

π11 π12 π13 π14

π23 π24

× f(π34 − π12; 1)f(π34 − π12; 2)f(π24 − π13; 0)f(π24 − π13; 1)

. . . . . .

.Example..

......

A P-partition π

π11 π12 π13 π14

π23 π24

× f(π34 − π12; 1)f(π34 − π12; 2)f(π24 − π13; 0)f(π24 − π13; 1)

. . . . . .

.. Okada’s (q, t)-hook formula conjecture

.Okada’s Conjecture..

......

Then the following identity would hold for any d-complete posets P:∑π∈A (P)

WP(π; q, t)zπ =∏v∈P

F (z[HP(v)])

where zπ =∏x∈P

zπ(x)c(x) . Here the sum on the left-hand side runs over

all P-partitions π ∈ A (P), and the right-hand side is the product ofall hook monomials for v ∈ P.

. . . . . .

Macdonald polynomials

. . . . . .

.. Arm-length and leg-length.Definition..

......

Let λ be a partition. Let s = (i, j) be a square in the diagram of λ, and leta(s) and l(s) be the arm-length and leg-length of s, i.e.,

a(s) = λi − j, l(s) = λ′j − i.

.Definition..

......

Define

bλ(q, t) :=∏s∈λ

1 − qa(s)t l(s)+1

1 − qa(s)+1t l(s)=

∏i≥1

f (λi − λi+m+1,m)

f (λi − λi+m,m),

belλ (q, t) :=

∏s∈λ

l(s) even

1 − qa(s)t l(s)+1

1 − qa(s)+1t l(s)=

∏i≥1

m≥0 even

f (λi − λi+m,m)

boaλ (q, t) :=

∏s∈λ

a(s) odd

bλ(s; q, t).

. . . . . .

.. Arm-length and leg-length.Definition..

......

Let λ be a partition. Let s = (i, j) be a square in the diagram of λ, and leta(s) and l(s) be the arm-length and leg-length of s, i.e.,

a(s) = λi − j, l(s) = λ′j − i.

.Definition..

......

Define

bλ(q, t) :=∏s∈λ

1 − qa(s)t l(s)+1

1 − qa(s)+1t l(s)=

∏i≥1

f (λi − λi+m,m),

belλ (q, t) :=

∏s∈λ

l(s) even

1 − qa(s)t l(s)+1

1 − qa(s)+1t l(s)=

∏i≥1

m≥0 even

f (λi − λi+m,m)

boaλ (q, t) :=

∏s∈λ

a(s) odd

bλ(s; q, t).

. . . . . .

.. Monomial symmeytric function

.Definition..

......

If x = (x1, x2, . . . ) and y = (y1, y2, . . . ) are two sequences of independentindeterminates, then we write

Π(x; y; q, t) =∏

(txiyj ; q)∞(xiyj ; q)∞

F(xiyj ; q, t).

.Definition..

......

Let Λn = Z[x1, . . . , xn]Sn and Λ denote the ring of symmetric polynomials

in n independent variables and the ring of symmetric polynomials incountably many variables, respectively. For λ = (λ1, . . . , λn) a partition ofat most n parts the monomial symmetric function mλ is defined as

mλ(x) :=∑α

where the sum is over all distinct permutations α of λ.

. . . . . .

.. Monomial symmeytric function

.Definition..

......

If x = (x1, x2, . . . ) and y = (y1, y2, . . . ) are two sequences of independentindeterminates, then we write

Π(x; y; q, t) =∏

(txiyj ; q)∞(xiyj ; q)∞

F(xiyj ; q, t).

.Definition..

......

Let Λn = Z[x1, . . . , xn]Sn and Λ denote the ring of symmetric polynomials

in n independent variables and the ring of symmetric polynomials incountably many variables, respectively. For λ = (λ1, . . . , λn) a partition ofat most n parts the monomial symmetric function mλ is defined as

mλ(x) :=∑α

where the sum is over all distinct permutations α of λ.

. . . . . .

.. Macdonald scalar product

.Definition..

......

For r a nonnegative integer the power sums pr are given by p0 = 1and pr = m(r) for r > 1. More generally the power-sum productsare defined as pλ(x) := pλ1(x)pλ2(x) · · · for an arbitrary partitionλ = (λ1, λ2, . . . ). Define the Macdonald scalar product ⟨·, ·⟩q,t onthe ring of symmetric functions by

⟨pλ, pµ⟩q,t := δλµzλ∏

n∏i=1

1 − qλi

1 − tλi

with zλ =∏

i≥1 imi mi! and mi = mi(λ).

. . . . . .

.. Macdonald’s P-function

.Definition..

......

If we denote the ring of symmetric functions in Λn variables over the fieldF = Q(q, t) of rational functions in q and t by Λn,F, then the Macdonaldpolynomial Pλ(x) = Pλ(x; q, t) is the unique symmetric polynomial in Λn,F

such that :Pλ =

∑µ≤λ

uλµ(q, t)mµ(x)

with uλλ = 1 and⟨Pλ,Pµ⟩q,t = 0 if λ , µ.

The Macdonald polynomials Pλ(x; q, t) with ℓ(λ) ≤ n form an F-basis ofΛn,F. If ℓ(λ) > n then Pλ(x; q, t) = 0. Pλ(x; q, t) is called Macdonald’sP-function. Since Pλ(x1, . . . , xn, 0; q, t) = Pλ(x1, . . . , xn; q, t) one canextend the Macdonald polynomials to symmetric functions containing aninfinite number of independent variables x = (x1, x2, . . . ), to obtain abasis of F = Λ ⊗ F.

. . . . . .

.. Macdonald’s Q-function

.Definition..

......

A second Macdonald symmetric function, called Macdonald’sQ-function, is defined as

Qλ(x; q, t) = bλ(q, t)Pλ(x; q, t).

The normalization of the Macdonald inner product is then⟨Pλ,Qµ⟩q,t = δλµ for all λ, µ, which is equivalent to∑

Pλ(x; q, t)Qλ(y; q, t) = Π(x; y; q, t).

. . . . . .

.. Pieri coefficients φλ/µ and ψλ/µ

.Definition..

......

Let r be a positive integer, and let λ, µ be partitions such that λ ⊃ µand λ − µ is a horizontal r-strip. The Pieri coefficients φλ/µ and ψλ/µare defined by

Pµgr =∑λ

φλ/µ Pλ,

Qµgr =∑λ

ψλ/µ Qλ,

where gr = Q(r).

. . . . . .

.. Another direct expression for ϕλ/µ and ψλ/µ

.Proposition..

......

From Macdonald’s book Chap.VI, §6, Ex.2(c), we have

φλ/µ(q, t) =∏

1≤i<j≤ℓ(λ)

f(λi − µj; j − i)f(µi − λj+1; j − i)

f(λi − λj; j − i)f(µi − µj+1; j − i),

ψλ/µ(q, t) =∏

1≤i<j≤ℓ(µ)

f(λi − µj; j − i)f(µi − λj+1; j − i)

f(µi − µj; j − i)f(λi − λj+1; j − i).

. . . . . .

.. Macdonald’s skew Q-function and skew P-function.Definition..

......

For any three partitions λ, µ, ν let fλµν be the coefficient Pλ in the productPµPν:

Pµ(x; q, t)Pν(x; q, t) =∑λ

fλµνPλ(x; q, t)

Now let λ, µ be partitions and define Qλ/µ ∈ ΛF by

Qλ/µ(x; q, t) =∑ν

fλµνQν(x; q, t).

Then Qλ/µ(x; q, t) = 0 unless λ ⊃ µ, and Qλ/µ is homogeneous of degree|λ| − |µ|, which is called Macdonald’s skew Q-function.

.Definition..

......

We define Macdonald’s skew P-function Pλ/µ by

Qλ/µ(x; q, t) =bλ(q, t)

bµ(q, t)Pλ/µ(x; q, t).

. . . . . .

.. Macdonald’s skew Q-function and skew P-function.Definition..

......

For any three partitions λ, µ, ν let fλµν be the coefficient Pλ in the productPµPν:

Pµ(x; q, t)Pν(x; q, t) =∑λ

fλµνPλ(x; q, t)

Now let λ, µ be partitions and define Qλ/µ ∈ ΛF by

Qλ/µ(x; q, t) =∑ν

fλµνQν(x; q, t).

Then Qλ/µ(x; q, t) = 0 unless λ ⊃ µ, and Qλ/µ is homogeneous of degree|λ| − |µ|, which is called Macdonald’s skew Q-function.

.Definition..

......

We define Macdonald’s skew P-function Pλ/µ by

Qλ/µ(x; q, t) =bλ(q, t)

bµ(q, t)Pλ/µ(x; q, t).

. . . . . .

.. Lemma.Lemma..

......

Let µ and ν be partitions, and x = (x1, x2, . . . ) and y = (y1, y2, . . . )are independent indeterminates.∑

Qλ/µ(x; q, t)Pλ/ν(y; q, t)

= Π(x; y; q, t)∑τ

Qν/τ(x; q, t)Pµ/τ(y; q, t)

Proof. ∑µ,ν

Qλ/µ(x)Pλ/ν(y)Qµ(z)Pν(w)

=∑µ,ν

Qλ(x, z)Pλ(y,w)

= Π(x, z; y,w)

= Π(x; y)Π(x;w)Π(z; y)Π(z;w)

. . . . . .

.. Lemma.Lemma..

......

= Π(x; y; q, t)∑τ

Proof. ∑µ,ν

=∑µ,ν

Qλ(x, z)Pλ(y,w)

= Π(x, z; y,w)

. . . . . .

.. Lemma.Lemma..

......

= Π(x; y; q, t)∑τ

Proof. ∑µ,ν

=∑µ,ν

Qλ(x, z)Pλ(y,w)

= Π(x, z; y,w)

. . . . . .

.. Lemma.Lemma..

......

= Π(x; y; q, t)∑τ

Proof. ∑µ,ν

=∑µ,ν

Qλ(x, z)Pλ(y,w)

= Π(x, z; y,w)

. . . . . .

.. Lemma.Lemma..

......

= Π(x; y; q, t)∑τ

Proof. ∑µ,ν

=∑µ,ν

Qλ(x, z)Pλ(y,w)

= Π(x, z; y,w)

. . . . . .

.. Proof

= Π(x; y)∑ξ

Qξ(x)Pξ(w)∑η

Qη(z)Pη(y)∑τ

Qτ(z)Pτ(w)

= Π(x; y)∑ξ,η,τ

Qξ(x)Pη(y)∑µ

bηbτbµ

fµητQµ(z)∑ν

fνξτPν(w)

= Π(x; y)∑µ,ν,τ

Qν/τ(x)Pµ/τ(y)Qµ(z)Pν(w)

Hence, by comparing the coefficients of Qµ(z)Pν(w) in the bothsides, we obtain the desired identity. This completes the proof.

. . . . . .

.. Proof

= Π(x; y)∑ξ

Qξ(x)Pξ(w)∑η

Qη(z)Pη(y)∑τ

Qτ(z)Pτ(w)

Qξ(x)Pη(y)∑µ

bηbτbµ

fµητQµ(z)∑ν

fνξτPν(w)

. . . . . .

.. Proof

= Π(x; y)∑ξ

Qξ(x)Pξ(w)∑η

Qη(z)Pη(y)∑τ

Qτ(z)Pτ(w)

Qξ(x)Pη(y)∑µ

bηbτbµ

fµητQµ(z)∑ν

fνξτPν(w)

. . . . . .

.. Proof

= Π(x; y)∑ξ

Qξ(x)Pξ(w)∑η

Qη(z)Pη(y)∑τ

Qτ(z)Pτ(w)

Qξ(x)Pη(y)∑µ

bηbτbµ

fµητQµ(z)∑ν

fνξτPν(w)

. . . . . .

.. A generalization of Vuletic’s formula

.Theorem..

......

Fix a positive integer T and two partitions µ0 and µT . Let x0, . . . , xT−1,y1, . . . , yT be sets of variables. Then we have

∑(λ1,µ1,λ2,...,λT )

T∏i=1

Qλi/µi−1(x i−1; q, t)Pλi/µi (y i ; q, t)

0≤i<j≤T

Π(x i , y j ; q, t)∑ν

QµT /ν(x0, . . . , xT−1; q, t)Pµ0/ν(y1, . . . , yT ; q, t)

where the sum runs over (2T − 1)-tuples (λ1, µ1, λ2, . . . , µT−1, λT ) ofpartitions satisfying

µ0 ⊂ λ1 ⊃ µ1 ⊂ λ2 ⊃ µ2 ⊂ · · · ⊃ µT−1 ⊂ λT ⊃ µT .

Proof. Use induction and the above lemma.

. . . . . .

.. A generalization of Vuletic’s formula

.Theorem..

......

Fix a positive integer T and two partitions µ0 and µT . Let x0, . . . , xT−1,y1, . . . , yT be sets of variables. Then we have

∑(λ1,µ1,λ2,...,λT )

T∏i=1

Qλi/µi−1(x i−1; q, t)Pλi/µi (y i ; q, t)

0≤i<j≤T

Π(x i , y j ; q, t)∑ν

QµT /ν(x0, . . . , xT−1; q, t)Pµ0/ν(y1, . . . , yT ; q, t)

where the sum runs over (2T − 1)-tuples (λ1, µ1, λ2, . . . , µT−1, λT ) ofpartitions satisfying

µ0 ⊂ λ1 ⊃ µ1 ⊂ λ2 ⊃ µ2 ⊂ · · · ⊃ µT−1 ⊂ λT ⊃ µT .

Proof. Use induction and the above lemma.

. . . . . .

.. A corollary

.Definition..

......

We define Pε[λ,µ]

(x; q, t) and Qε[λ,µ]

(x; q, t) for a pair (λ, µ) of partitions, aset x = (x1, x2, . . . ) of independent variables and ε = ± by

Pε[λ,µ](x; q, t) =

Pλ/µ(x; q, t) if ε = +,Qµ/λ(x; q, t) if ε = −,

Qε[λ,µ](q, t) =

Qλ/µ(x; q, t) if ε = +,Pµ/λ(x; q, t) if ε = −.

Here we assume λ ⊃ µ if ε = +, and λ ⊂ µ if ε = −.

. . . . . .

.. A corollary

.Theorem..

......

Let n be a positive integer, and ϵ = (ϵ1, . . . , ϵn) a sequence of ±, Fix apositive integer n and two partitions λ0 and λn. Let x1, . . . , xn be sets ofvariables. Then we have∑

(λ1,λ2,...,λn−1)

n∏i=1

Pϵi[λi−1,λi ]

(x i ; q, t) =∏

i<j(ϵi ,ϵj )=(−,+)

Π(x i ; x j ; q, t)

×∑ν

Qλn/ν({x i}ϵi=−; q, t)Pλ0/ν({x i}ϵi=+; q, t),

where the sum runs over (n − 1)-tuples (λ1, λ2, . . . , λn−1) of partitionssatisfying λi−1 ⊃ λi if ϵi = +,

λi−1 ⊂ λi if ϵi = −.

Proof. Take T = n and put X i−1 = 0 and Y i = x i if ϵi = +1, andX i−1 = x i and Y i = 0 if ϵi = −1.

. . . . . .

.. A corollary

.Theorem..

......

(λ1,λ2,...,λn−1)

n∏i=1

Pϵi[λi−1,λi ]

(x i ; q, t) =∏

i<j(ϵi ,ϵj )=(−,+)

Π(x i ; x j ; q, t)

×∑ν

. . . . . .

.. A corollary

.Theorem..

......

(λ1,λ2,...,λn−1)

n∏i=1

Pϵi[λi−1,λi ]

(x i ; q, t) =∏

i<j(ϵi ,ϵj )=(−,+)

Π(x i ; x j ; q, t)

×∑ν

. . . . . .

.. Notation

.Definition..

......

Under the assumption that λ ⊇ µ if ε = −, or λ ⊆ µ if ε = +, we write

Ψελ/µ =

ψλ/µ if ε = −,φµ/λ if ε = +,

Φελ/µ =

φλ/µ if ε = −,ψµ/λ if ε = +.

.Definition..

......

Here we assume λ ≻ µ if δ = +1, and λ ≺ µ if δ = −1. We also write

|λ − µ|δ =|λ − µ| if δ = +1,|µ − λ| if δ = −1.

. . . . . .

.. Notation

.Definition..

......

Ψελ/µ =

Φελ/µ =

.Definition..

......

Here we assume λ ≻ µ if δ = +1, and λ ≺ µ if δ = −1. We also write

|λ − µ|δ =|λ − µ| if δ = +1,|µ − λ| if δ = −1.

. . . . . .

.. Notation

.Definition..

......

Let n be a positive integer. Let ϵ = (ϵ1, . . . , ϵn) be a sequence of ±1. Let(λ0, λ1, . . . , λn) be an (n + 1)-tuple of partitions such that λi−1 ≻ λi ifϵ = +1, and λi−1 ≺ λi if ϵ = −1. Then we write

ϕϵ[λ0,λ1,...,λn](q, t) =n∏

ϕϵi[λi−1,λi ]

(q, t), ψϵ[λ0,λ1,...,λn](q, t) =n∏

ψϵi[λi−1,λi ]

(q, t).

.Definition..

......

Let α be a strict partition, and let n be an integer such that n ≥ α1. Definea sequence ϵn(α) = (ϵ1(α), . . . , ϵn(α)) of ±1 by putting

ϵk (α) =

+1 if k is a part of α,−1 if k is not a part of α.

. . . . . .

.. Notation

.Definition..

......

Let n be a positive integer. Let ϵ = (ϵ1, . . . , ϵn) be a sequence of ±1. Let(λ0, λ1, . . . , λn) be an (n + 1)-tuple of partitions such that λi−1 ≻ λi ifϵ = +1, and λi−1 ≺ λi if ϵ = −1. Then we write

ϕϵ[λ0,λ1,...,λn](q, t) =n∏

ϕϵi[λi−1,λi ]

(q, t), ψϵ[λ0,λ1,...,λn](q, t) =n∏

ψϵi[λi−1,λi ]

(q, t).

.Definition..

......

Let α be a strict partition, and let n be an integer such that n ≥ α1. Definea sequence ϵn(α) = (ϵ1(α), . . . , ϵn(α)) of ±1 by putting

ϵk (α) =

+1 if k is a part of α,−1 if k is not a part of α.

. . . . . .

.. Example of ϵ(α) and k th trace π[k ].Definition..

......

For each integer k = 0, . . . , n we define the k th trace π[k ] to be thesequence (. . . , π2,k+2, π1,k+1) obtained by reading the k th diagonal fromSE to NW. Here we use the convention that π[k ] = ∅ if k ≥ α1.

.Example..

......

For example, if α = (8, 5, 2, 1) and n = 10, then we haveϵ = (+ + − −+ − −+ −−).

π11 π12 π13 π14 π15 π16 π17 π18

π22 π23 π24 π25 π26

π33 π34

We have π[0] = (π44, π33, π22, π11), π[1] = (π34, π23, π12),. . . , π[10] = ∅,

. . . . . .

.. Example of ϵ(α) and k th trace π[k ].Definition..

......

For each integer k = 0, . . . , n we define the k th trace π[k ] to be thesequence (. . . , π2,k+2, π1,k+1) obtained by reading the k th diagonal fromSE to NW. Here we use the convention that π[k ] = ∅ if k ≥ α1.

.Example..

......

For example, if α = (8, 5, 2, 1) and n = 10, then we haveϵ = (+ + − −+ − −+ −−).

π11 π12 π13 π14 π15 π16 π17 π18

π22 π23 π24 π25 π26

π33 π34

We have π[0] = (π44, π33, π22, π11), π[1] = (π34, π23, π12),. . . , π[10] = ∅,

. . . . . .

Tailed Inset Case

. . . . . .

.. Tailed Insets

.Definition..

......

α1 > α2 > α3 ≥ 0, β1 > β2 ≥ 0.

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Tailed Insets

.Definition..

......

α1 > α2 > α3 ≥ 0, β1 > β2 ≥ 0.

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Tailed Insets

.Definition..

......

α1 > α2 > α3 ≥ 0, β1 > β2 ≥ 0.

(i1, j1)≥(i2, j2)⇐⇒ i1 ≤ i2 and j1 ≤ j2.

. . . . . .

.. Tailed Insets

(1, 1)

. . . . . .

.. Tailed Insets

−4 −3 −2 −1 0 1 2 3 4 5 6 7

−1′

+ +−

. . . . . .

.. P-partition for Tailed Insets

.Definition..

......

Let π = (σ, τ, ρ, γ, δ) ∈ A (P) be a P-partition as in the followingfigure.

ρ4 ρ3 ρ2 ρ1 σ11 σ12 σ13 σ14 σ15 σ16 σ17 σ18

γ σ22 σ23 σ24 σ25 σ26

τ21 σ33 σ34 σ35

τ31 τ32 δ

τ41 τ42

. . . . . .

.. Banner partition

.Definition..

......

Let pi (resp. qi) denote the number of vertices in the ith diagonal of λ(resp. µ) for i ≥ 1, whereas we set p0 = 3 and q0 = 2. We defineε = (εc,c+1)c∈Z as follows. If c ≥ 1,

εc,c+1 =

+ if pc = pc−1,− if pc = pc−1 − 1,

and if c ≤ 0,

εc,c+1 =

− if q−c+1 = q−c ,+ if q−c+1 = q−c − 1.

The color of each vetex is shown in the figure above. In this example, wehave (pi)i≥1 = (332211100 . . . ), (qi)i≥1 = (221100 . . . ) and p0 = 3,q0 = 2 by definition. Hence we haveελ = (· · · − −+ −+ − −++ −+ −++ −++ · · · ) as in the above figure.

. . . . . .

.. (q, t)-weight.Definition..

......

If we set

Wc,dN (π; q, t) =

∏x,y∈P, x<y

c(x) = c and c(y) = d

(π(x) − π(y);

⌊r(y) − r(x)

if c and d are adjacent colors in T , and

Wc,+D (π; q, t) =

∏x,y∈P, x<y

c(x)=c(y)=c

(π(x) − π(y),

⌊r(y) − r(x)

Wc,−D (π; q, t) =

∏x,y∈P, x<y

c(x)=c(y)=c

f(π(x) − π(y);

⌊r(y) − r(x)

⌋− 1

then we have WP(π; q, t) =

∏c and d are adjacent in T Wc,d

N (π; q, t)∏c all colors in T Wc

D(π; q, t). where

WcD(π; q, t) = Wc,+

D (π; q, t)Wc,−D (π; q, t).

. . . . . .

.. WP(π; q, t) for Banners

.Definition..

......

If λ and µ are partitions such that λ − µ is a horizontal strip, then it isknown that

ψλ/µ(q, t) =∏

1≤i≤j≤ℓ(µ)

fq,t(λi − µj ; j − i)fq,t(µi − λj+1; j − i)

fq,t(µi − µj ; j − i)fq,t(λi − λj+1; j − i),

φλ/µ(q, t) =∏

1≤i≤j≤ℓ(λ)

fq,t(λi − µj ; j − i)fq,t(µi − λj+1; j − i)

fq,t(λi − λj ; j − i)fq,t(µi − µj+1; j − i).

Ψελ/µ =

Φελ/µ =

. . . . . .

.Definition..

......

I) For 0 ≤ c ≤ λ1, we define the partition Λc of length ≤ pc by

Λc = (σpc ,pc+c , . . . , σ1,1+c) = (σpc+1−i,pc+1−i+c)1≤i≤pc .

II) Now we setΛ−1 = (τq1+1,q1 , . . . , τ2,1︸︷︷︸

, γ, ρ1),

where q1 = 1 or 2.

III) If −µ1 ≤ c ≤ −2, then we set

Λc = (τq−c−c,q−c , . . . , τ1−c,1︸︷︷︸q−c

, γ, . . . , γ︸︷︷︸−c

, ρ−c),

where q−c = 1 or 2.

IV) If c = −µ1 − 1, then we set

Λ−µ1−1 = (γ, γ, . . . , γ︸︷︷︸µ1+1

. . . . . .

.Definition..

......

, γ, ρ1),

where q1 = 1 or 2.

Λc = (τq−c−c,q−c , . . . , τ1−c,1︸︷︷︸q−c

, γ, . . . , γ︸︷︷︸−c

, ρ−c),

Λ−µ1−1 = (γ, γ, . . . , γ︸︷︷︸µ1+1

. . . . . .

.Definition..

......

, γ, ρ1),

where q1 = 1 or 2.

Λc = (τq−c−c,q−c , . . . , τ1−c,1︸︷︷︸q−c

, γ, . . . , γ︸︷︷︸−c

, ρ−c),

Λ−µ1−1 = (γ, γ, . . . , γ︸︷︷︸µ1+1

. . . . . .

.Definition..

......

, γ, ρ1),

where q1 = 1 or 2.

Λc = (τq−c−c,q−c , . . . , τ1−c,1︸︷︷︸q−c

, γ, . . . , γ︸︷︷︸−c

, ρ−c),

Λ−µ1−1 = (γ, γ, . . . , γ︸︷︷︸µ1+1

. . . . . .

.Definition..

......

, γ, ρ1),

where q1 = 1 or 2.

Λc = (τq−c−c,q−c , . . . , τ1−c,1︸︷︷︸q−c

, γ, . . . , γ︸︷︷︸−c

, ρ−c),

Λ−µ1−1 = (γ, γ, . . . , γ︸︷︷︸µ1+1

. . . . . .

.. (q, t)-weight by Pieri coefficient.Theorem..

......

If P = P5(λ, µ) is the Tailed Insets corresponding to strict partitions λ andµ, then we have

WP(π; q, t) =fq,t(γ; 0)

∏3i=1 fq,t(δ − σi,i ; 3 − i)

fq,t(δ − γ; 2)fq,t(δ − γ; 1)

λ1∏c=−µ1−1

Ψεc,c+1

Λc/Λc+1.

.Proposition..

......

We set

Zc =c∏

k=−µ1−1

zk , Zc.d =Zd

d∏k=c+1

where z−µ1−1 is a dummy variable which does not appear in the original

weight. Then we have zπ =zγ+δ−1′

−1∏c=−µ1−1

·λ1∏

c=−µ1−1

Z |Λc |−|Λc+1 |

. . . . . .

.. (q, t)-weight by Pieri coefficient.Theorem..

......

If P = P5(λ, µ) is the Tailed Insets corresponding to strict partitions λ andµ, then we have

WP(π; q, t) =fq,t(γ; 0)

∏3i=1 fq,t(δ − σi,i ; 3 − i)

fq,t(δ − γ; 2)fq,t(δ − γ; 1)

λ1∏c=−µ1−1

Ψεc,c+1

Λc/Λc+1.

.Proposition..

......

We set

Zc =c∏

k=−µ1−1

zk , Zc.d =Zd

d∏k=c+1

where z−µ1−1 is a dummy variable which does not appear in the original

weight. Then we have zπ =zγ+δ−1′

−1∏c=−µ1−1

·λ1∏

c=−µ1−1

Z |Λc |−|Λc+1 |

. . . . . .

.. Fix Certain Colors

.Definition..

......

We use the convention that ε−µ1−1,−µ1 = + and εc,c+1 = − for c < −µ1 − 1.Note that ♯{c < 0 | εc,c+1 = +} = 2. Because ε−µ1−1,−µ1 = + andεc,c+1 = − for c < −µ1 − 1, we may set

{c < 0 | εc,c+1 = +} = {c−1 , c−2 }.

where −µ1 − 1 = c−2 < c−1 < 0 Also note that ♯{c ≥ 0 | εc,c+1 = −} = 3.Because ελ1,λ1+1 = − and εc,c+1 = + for c > λ1, we may set

{c ≥ 0 | εc,c+1 = −} = {c+1 , c

+2 , c

where 0 ≤ c+1 < c+

2 < c+3 = λ1. Hence we have

−µ1 − 1 = c−2 < c−1 < 0 ≤ c+1 < c+

2 < c+3 = λ1.

. . . . . .

.. Left-Hand Side.Theorem..

......

∑WP(π; q, t)zπ =

∏0≤i<j

εi,i+1=+εj,j+1=−

Π(Z−1i ;Zj ; q, t)

∏i<j<0

εi,i+1=+εj,j+1=−

Π(Z−1i ;Zj ; q, t)

σ1,1 ,σ2,2 ,σ3,3γ,δ,ν3

fq,t(γ; 0)∏3

i=1 fq,t(δ − σi,i ; 3 − i)

fq,t(δ − γ; 2)fq,t(δ − γ; 1)·

zγ+δ−1′∏−1c=−µ1−1 Zγ

× PΛ0(Zc+2,Zc+

3,Zλ1 ; q, t)

× QΛ0/ν(Z−1−µ1−1,Z

−1c−1; q, t)PΛ−µ1−1/ν(Z−µ1 , . . . , Zc−1 , · · ·Z−1; q, t),

where the sum runs over

0 ≤ ν3 ≤ σ1,1 ≤ γ ≤ σ2,2 ≤ σ3,3 ≤ δ

and Λ0 = (σ3,3, σ2,2, σ1,1), Λ−µ1−1 = (γ, γ, . . . , γ︸︷︷︸µ1+1

) and ν = (γ, γ, ν3).

. . . . . .

.. Right-hand side

.Definition..

......

We put P = P1 ∪ P2 ∪ P3 ∪ P4 ∪ P5 ∪ P6, where

P1 ={(i, i + c) | j > 3, 1 ≤ i ≤ pi , 1 ≤ c ≤ λ1,

{(j + 1 − c, j + 1) | i > 3, 1 ≤ j ≤ qc , −µ1 ≤ c ≤ −1

P3 ={(i, j) | 1 ≤ i ≤ 3, 2 ≤ j ≤ 3

P4 ={(2, 1), (1, 1), (1, 0)

P5 ={(1, c + 1) | − µ1 ≤ c ≤ −2

P6 ={(4, 4)

}and we write

Ri =∏v∈Pi

F (z [HP(v)] ; q, t) ,

for i = 1, . . . , 6.

. . . . . .

.. Right-hand side

(1,−3) . . . (1, 1) (1, 2) (1, 3) . . .

(1, 8)

(2, 1)

(6, 2)

(4, 4)

+ +−

+ + . . .

−−

c−2 = −5, c−1 = −3.

c+1 = 2, c+

2 = 4, c+3 = 7.

. . . . . .

.. Interpretation of RHS.Proposition..

......

By direct computation, is is not hard to see

R1 =∏0≤i<j

εi,i+1=+εj,j+1=−

F (Zi,j; q, t) ,

R2 =∏i<j<0

εi,i+1=+εj,j+1=−

F (Zi,j; q, t) .

R3 =∏i<0≤j

εi,i+1=+εj,j+1=−

F (wZi,j; q, t)

= F(wZc−1 ,c

+1; q, t

(wZc−1 ,c

+2; q, t

(wZc−1 ,c

+3; q, t

(wZc−2 ,c

+1; q, t

(wZc−2 ,c

+2; q, t

(wZc−2 ,c

+3; q, t

)Masao Ishikawa (q, t)-hook formula for Tailed Insets

. . . . . .

.. Interpretation of RHS

.Proposition..

......

R4 = F(wZc−2 ,c

Zc−1 ,c+2; q, t

(wZc−2 ,c

Zc−1 ,c+3; q, t

(wZc−2 ,c

Zc−1 ,c+2; q, t

P5 =−1∏

c=−µ1c,c−1

F(w2Zc−2 ,c

Zc−1 ,c+2

Zc,c+3; q, t

. . . . . .

.. The end

Thank you for your attention!

)-hook formula for tailed insets and a macdonald ...mi/comb2018/talks/feb23/ishikawa201… ·...

Documents